Theorem mobii 2114

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2113	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1404	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 105  ⊤wtru 1396  ∃*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-eu 2080  df-mo 2081

This theorem is referenced by:  moaneu  2154  moanmo  2155  2moswapdc  2168  2exeu  2170  rmobiia  2722  rmov  2820  euxfr2dc  2988  rmoan  3003  2rmorex  3009  mosn  3702  dffun9  5347  funopab  5353  funco  5358  funcnv2  5381  funcnv  5382  fun2cnv  5385  fncnv  5387  imadif  5401  fnres  5440  ovi3  6148  oprabex3  6280  axaddf  8066  axmulf  8067  frecuzrdgtcl  10646  frecuzrdgfunlem  10653  fsum3  11913